Probability-Preserving Zonotopic Neural Uncertainty

1. Exact integration point: implement a parallel uncertainty wrapper around an MLP, small RNN, or state-space model. The nominal input is accompanied by a finite mixture of zonotopes representing sensor modes, quantization modes, bounded perturbation regimes, or multiple posterior hypotheses. Each mixture component is propagated layer by layer. For affine layers use center = W @ center + b and generators = W @ generators. For ReLU, compute coordinatewise lower and upper bounds from the current zonotope using center - sum(abs(generators), axis=1) and center + sum(abs(generators), axis=1). Apply the exact zero or identity transformation when the interval is one-sided; for crossing intervals, use a conservative affine relaxation and add a nonnegative generator for the relaxation gap. 2. Pseudocode: initialize components (alpha_i,c_i,E_i); for each layer, propagate every component; after propagation, rank pairs by center distance and zonotope volume; propose clusters; compute an enclosing zonotope for each cluster; accept a merge only if the enclosure contains every source zonotope, then replace the cluster by (sum(alpha_i), c_tilde, E_tilde). Keep the top K components only if discarded mass is reported rather than silently renormalized. 3. Computed from the paper: mixture-weight addition and the set-containment condition are exact. Estimated empirically: enclosure tightness, volume proxies, and ReLU relaxation error. A simple enclosure uses the box hull of all source zonotopes, with center equal to the midpoint of coordinatewise extrema and diagonal generators equal to half-widths; this is conservative and easy to verify. 4. First cheap experiment: use a two-layer ReLU MLP on MNIST or two-moons. Create two input-noise modes with weights 0.7 and 0.3, each a bounded box or zonotope, and compare one-box interval propagation with mixture propagation at budgets K=1,2,4,8. Measure certified class stability and empirical coverage over 10,000 sampled perturbations. The primary prediction is zero false-coverage violations whenever affine and ReLU relaxations are conservative; after admissible reduction, empirical coverage of the retained region must be at least the retained probability mass, up to numerical tolerance. A second prediction is that merging separated modes into one box sharply increases reachable-set volume, whereas keeping them separate preserves multimodal predictions. Plot certified volume versus component budget and identify the knee where coverage improves most per component.